No direct knowledge but I would caution you not to expect to have the pick of good tutors as I would think that most would have been booked up some time ago (if it is anything like the tutoring scene in NW London).

I'm afraid that Writing is probably right. In NW Londin, there may be some changeover in January, after private school examinations, but not much beyond that until the September (state school 11+ exams in NW London). Many tutees also choose to stay on a further year until they start secondary school in September so that they're still 'strong' and do well in CAT tests used to set children according to ability. A good tutor will inevitably have a waiting list. However, many of those on the waiting list will/ won't be able to take the space that becomes available on a particular day and at a particular time, as it will depend on their own/ their siblings' other commitments. It's worth adding yourself onto a waiting list and hoping for the best. I hope it's easier in your area.

We didn't have such issues on timing. I started looking in September, found one and did 10 sessions pre-exam in January.

The trick was to find a really local tutor. If you live walking distance from them, they are much more flexible on timings and availability. It gets much harder for the "uber tutors" when they have to travel all over London every evening.

Use your post code or village name in Google to spot them. Local ads in newsagents too. Obviously, you will want to vet them and usually a phone conversation will give you a good "gut" feeling.